The level crossings of random sums

TL;DR

This paper derives an exact formula for the expected density of complex zeros of random sums involving Gaussian variables and holomorphic functions, linking the problem to Brownian motion observations.

Contribution

It provides a novel exact formula for the expected density of complex zeros of random sums with Gaussian coefficients and explores their relation to Brownian motion.

Findings

Derived an exact formula for the expected density of complex zeros.

Reformulated the problem in terms of Brownian motion observations.

Analyzed the expected number of zeros for nonvanishing mean coefficients.

Abstract

Let $\{\eta_{j}\}_{j = 0}^{N}$ be a sequence of independent and identically distributed complex normal random variables with mean zero and variances $\{\sigma_{j}^{2}\}_{j = 0}^{N}$. Let $\{f_{j} (z)\}_{j = 0}^{N}$ be a sequence of holomorphic functions that are real-valued on the real line. The purpose of the present study is that of examining the number of times that the random sum $\sum_{j = 0}^{N} \eta_{j} f_{j} (z)$ crosses the complex level $\boldsymbol{K} = K_{1} + i K_{2}$, where $K_{1}$ and $K_{2}$ are constants independent of $z$. More specifically, we establish an exact formula for the expected density function for the complex zeros. We then reformulate the problem in terms of successive observations of a Brownian motion. We further answer the basic question about the expected number of complex zeros for coefficients of nonvanishing mean values.